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Gromov hyperbolicity in Cartesian product graphs

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Abstract

If X is a geodesic metric space and x 1, x 2, x 3 ∈ X, a geodesic triangle T = {x 1, x 2, x 3} is the union of the three geodesics [x 1 x 2], [x 2 x 3] and [x 3 x 1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e. δ(X) = inf{δ ≥ 0: X is δ-hyperbolic}. In this paper we characterize the product graphs G 1 × G 2 which are hyperbolic, in terms of G 1 and G 2: the product graph G 1 × G 2 is hyperbolic if and only if G 1 is hyperbolic and G 2 is bounded or G 2 is hyperbolic and G 1 is bounded. We also prove some sharp relations between the hyperbolicity constant of G 1 × G 2, δ(G 1), δ(G 2) and the diameters of G 1 and G 2 (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the precise value of the hyperbolicity constant for many product graphs.

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References

  1. Balogh Z M and Buckley S M, Geometric characterizations of Gromov hyperbolicity, Invent. Math. 153 (2003) 261–301

    Article  MATH  MathSciNet  Google Scholar 

  2. Bermudo S, Rodríguez J M, Sigarreta J M and Vilaire J-M, Mathematical properties of Gromov hyperbolic graphs, to appear in AIP Conference Proceedings (2010)

  3. Bermudo S, Rodríguez J M, Sigarreta J M and Vilaire J-M, Gromov hyperbolic graphs, preprint

  4. Bermudo S, Rodríguez J M, Sigarreta J M and Tourís E, Hyperbolicity and complement of graphs, preprint

  5. Bonk M, Heinonen J and Koskela P, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001) 99 pp.

    Google Scholar 

  6. Bowditch B H, Notes on Gromov’s hyperobolicity criterion for path-metric spaces. Group theory from a geometrical viewpoint, Trieste, 1990 (eds) E Ghys, A Haefliger and A Verjovsky (River Edge, NJ: World Scientific) (1991) pp. 64–167

    Google Scholar 

  7. Brinkmann G, Koolen J and Moulton V, On the hyperbolicity of chordal graphs, Ann. Comb. 5 (2001) 61–69

    Article  MATH  MathSciNet  Google Scholar 

  8. Chepoi V, Dragan F F, Estellon B, Habib M and Vaxes Y, Notes on diameters, centers and approximating trees of δ-hyperbolic geodesic spaces and graphs, Electr. Notes Discrete Math. 31 (2008) 231–234

    Article  MathSciNet  Google Scholar 

  9. Frigerio R and Sisto A, Characterizing hyperbolic spaces and real trees, Geom Dedicata 142 (2009) 139–149

    Article  MATH  MathSciNet  Google Scholar 

  10. Ghys E and de la Harpe P, Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83 (Boston, MA: Birkhauser Boston Inc.) (1990)

    MATH  Google Scholar 

  11. Gromov M, Hyperbolic groups, in ‘Essays in group theory’ (ed) S M Gersten (MSRI Publ., Springer) (1987) vol. 8, pp. 75–263

  12. Gromov M (with appendices by M Katz, P Pansu and S Semmes), Metric structures for Riemannian and non-Riemannnian spaces, Progress in Mathematics (Birkhäuser) (1999) vol. 152

  13. Hästö P A, Gromov hyperbolicity of the j G and \(\tilde J_G\) metrics, Proc. Am. Math. Soc. 134 (2006) 1137–1142

    Article  MATH  Google Scholar 

  14. Hästö P A, Lindén H, Portilla A, Rodríguez J M and Tourís E, Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics, preprint

  15. Hästö P A, Portilla A, Rodríguez J M and Tourís E, Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains, to appear in Bull. London Math. Soc.

  16. Hästö P A, Portilla A, Rodríguez J M and Tourís E, Comparative Gromov hyperbolicity results for the hyperbolic and quasihyperbolic metrics, to appear in Complex Variables and Elliptic Equations

  17. Hästö P A, Portilla A, Rodríguez JM and Tourís E, Uniformly separated sets and Gromov hyperbolicity of domains with the quasihyperbolic metric, to appear in Medit. J. Math.

  18. Jonckheere E A, Controle du trafic sur les reseaux a geometrie hyperbolique-Une approche mathematique a la securite de l’acheminement de l’information, Journal Europeen de Systemes Automatises 37(2) (2003) 145–159

    Article  MathSciNet  Google Scholar 

  19. Jonckheere E A and Lohsoonthorn P, Geometry of network security, American Control Conf. ACC (2004) 111–151

    Google Scholar 

  20. Jonckheere E A, Lohsoonthorn P and Bonahon F, Scaled Gromov hyperbolic graphs, J. Graph Theory 2 (2007) 157–180

    MathSciNet  Google Scholar 

  21. Koolen J H and Moulton V, Hyperbolic Bridged Graphs, Europ. J. Combinatorics 23 (2002) 683–699

    Article  MATH  MathSciNet  Google Scholar 

  22. Michel J, Rodríguez J M, Sigarreta J M and Villeta M, Hyperbolicity and parameters of graphs, to appear in Ars Combinatoria

  23. Oshika K, Discrete groups (AMS Bookstore) (2002)

  24. Portilla A, Rodríguez J M and Tourís E, Gromov hyperbolicity through decomposition of metric spaces II, J. Geom. Anal. 14 (2004) 123–149

    MATH  MathSciNet  Google Scholar 

  25. Portilla A and Tourís E, A characterization of Gromov hyperbolicity of surfaces with variable negative curvature, Publ. Mat. 53 (2009) 83–110

    MATH  MathSciNet  Google Scholar 

  26. Rodríguez J M, Sigarreta J M, Vilaire J-M and Villeta M, On the hyperbolicity constant in graphs, preprint

  27. Rodríguez J M and Tourís E, Gromov hyperbolicity through decomposition of metric spaces, Acta Math. Hungar. 103 (2004) 53–84

    Article  Google Scholar 

  28. Rodríguez J M and Tourís E, A new characterization of Gromov hyperbolicity for Riemann surfaces, Publ. Mat. 50 (2006) 249–278

    MATH  MathSciNet  Google Scholar 

  29. Rodríguez J M and Tourís E, Gromov hyperbolicity of Riemann surfaces, Acta Math. Sinica 23 (2007) 209–228

    Article  MATH  Google Scholar 

  30. Tourís E, Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces, preprint

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Av. de la Universidad 30, 28911, Leganés, Madrid, Spain

    Junior Michel & José M. Rodríguez

  2. Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, México

    José M. Sigarreta

  3. Departamento de Estadística e Investigación Operativa III, Universidad Complutense de Madrid, Av. Puerta de Hierro s/n., 28040, Madrid, Spain

    María Villeta

Authors
  1. Junior Michel
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  2. José M. Rodríguez
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  3. José M. Sigarreta
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  4. María Villeta
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Correspondence to Junior Michel.

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Michel, J., Rodríguez, J.M., Sigarreta, J.M. et al. Gromov hyperbolicity in Cartesian product graphs. Proc Math Sci 120, 593–609 (2010). https://doi.org/10.1007/s12044-010-0048-6

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  • DOI: https://doi.org/10.1007/s12044-010-0048-6

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